Coin flip 2 Players

Question

Player 1 and Player 2 are playing a game of flipping a coin 5 times. The player with the most heads win.

What is the probability of Player 1 flipping 3 heads and 2 tails and Play 2 flipping 2 heads and 3 tails?

Answer 1

The first flip of player 1, there are 2 possibilities. The second flip of player 1 there are 2 so now, total, there are 2•2 or 4 possibilities. This continues so that the total number of possibilities for player 1 and player 2 each flipping 5 times is:

2•2•2•2•2•2•2•2•2•2 = 2¹⁰ = 1024 possibilities

Now, counting the possibilities where Player 1 has 3H/2T and Player 2 has 2H/3T:

Player 1

Scenarios with 3H and 2T:

H H H T T

H H T H T

H H T T H

H T H H T

H T H T H

H T T H H

T H H H T

T H H T H

T H T H H

T T H H H

Player 2

Scenarios with 2H and 3T:

T T T H H

T T H T H

T T H H T

T H T T H

T H T H T

T H H T T

H T T T H

H T T H T

H T H T T

H H T T T

That's 20 possibilities out of the 1024 total possibilities so the probability is 20/1024 = 5/256